which can be easily checked from (REF ) or by
writing the theta functions in terms of Dedekind eta
function, see [1]}.
Applying \(\tau \)  derivatives on both sides and using Darboux–Halphen equations, one can also check that the solution to Darboux–Halphen system (REF ) are the roots of the cubic equation
\(y^3-\frac{3}{2}\gamma (\tau )y^2+\frac{3}{2}\gamma ^{\prime }(\tau )y -\frac{1}{4}\gamma ^{\prime \prime }(\tau )=0.\) 
